FINA 1082 Financial Management

Transcription

1 FINA 1082 Financial Management Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA257 Department of Accounting and Finance

2 Lecture 1 Introduction to Business Finance Introduction to Financial Economics September, 28,

3 Subject Objectives Solve basic problems in financial mathematics Discuss the basic theories underlying the pricing of risky assets Comprehend the concepts of portfolio formation and risk diversification Explain the fundamentals of capital budgeting, including the use of alternative criteria, allowing for inflation and the treatment of risk Analyze the issues facing managers in capital structure and dividend policy decisions Use the features of financial derivatives to achieve specific financial outcomes. 3

4 Recommended Text Brealey R., Myers S., and Allen F., Principles of Corporate Finance, 8 th Edition, McGraw-Hill/Irwin,

5 Introduction to Business Finance Overview of the Finance discipline Compare simple interest to compounded interest Compute the future value of a single cash flow Compute the present value of a single cash flow Compute an unknown interest rate and time period 5

6 What is Finance? Finance is the study of how individuals, businesses and institutions acquire, spend and manage financial resources Major areas of Finance Investment analysis and management Corporate Finance Capital markets and Financial Institutions International Finance Personal finance Real Estate Finance This subject provides an introduction to Investment Analysis and Corporate Finance 6

7 Overview of Business Finance The study of Finance is related to the corporate objective of maximizing shareholder wealth Our focus is on financial decision making Individuals/Investors Financial Security Valuation: Earnings and Dividend models Portfolios and Risk Diversification: Portfolio Analysis Determination of Security Prices and Rates of Return: Capital Asset Pricing Model and Arbitrage Pricing Model Using Financial Derivatives: Futures, Forwards and Options Financial Managers Investment Decisions: Capital Budgeting Analysis Financing Decisions: Capital Structure and Dividend Policies.and the interaction among these decisions 7

8 This Subject Investment Analysis is mainly concerned with where and how to invest Valuation of stocks, bonds and derivatives Portfolio Diversification Asset Pricing and Market efficiency These topics (except derivatives) are covered in the first half of this subject, Corporate Finance is mainly concerned with the decisions of managers Capital Budgeting What investments to make Capital Structure How to finance these investments Dividend Policy What to payout to Shareholders These topics (along with derivatives) are covered in the second half of this subject. 8

9 Why Study Finance To make informed economic decisions To better manage existing financial resources and accumulate wealth over time To be successful in the business world you need to have an understanding of finance. 9

10 The Finance Function The main goal of managers is to maximize the market value of the firm Value of the firm = Present value of Future expected cash flows This maximizes the wealth of shareholders Shareholder wealth = Present Value of shareholder s future expected cash flows Do managers always maximize firm value? What about these? HIH Insurance and OneTel collapses (Australia) Enron and Worldcom collapses (USA) A qualified YES? 10

11 The Market Value of the Firm Firm Value is the Present Value of the Future expected cash flows Firm Value = n t=1 E CF 1+k E(CF t ) = Expected cash flows received at the end of period t n = Number of periods over which cash flows are received k = required rate of returns by investors Main factors to consider when valuing a firm: Magnitude of expected cash flows E (CF t ) Timing of cash flows n Risk of Expected Cash Flows k Efficiency of capital markets t t 11

12 Introduction to Financial Mathematics 1 Simple Interest The value of a cash flow is calculated without including any accrued interest to the principal Example: If you invest $1,000 at 8% p.a. earning simple interest for 5 years what amount will you have in your account at the end of that time period? Interest earned in each of the five years = = $80 Interest earned in over five years = 1000 (5 0.08) = $400 Future value at the end of year 5 = = $1,400 Future value at the end of year 5 = 1000 ( ) = $1,400 Future value (Simple interest): Sn = P0 (1 + n i) Present value (Simple interest): P0 = Sn / (1 + n i) 12

13 Simple Versus Compounded Interest Compound Interest Interest accrued is added to the principal The value of a cash flow is calculated based on the principal and interest accrued Example: If you invest $1,000 at 8% p.a. earning compounded interest for 5 years what amount will you have in your account at the end of that time period? Future value at the end of year 1 = 1000 (1.08) = $1, Future value at the end of year 2 = 1080 (1.08) = $1, Future value at the end of year 5 = 1000 (1.08) 5 = $1, The difference of $69.33 (= ) is due to the compounding of interest 13

14 Simple Versus Compounded Interest Amount Invested $1,000 Interest Rate 8% End of year Simple Interest Compounded Interest Difference 1 $1, $1, $ $1, $1, $ $1, $1, $ $1, $1, $ $1, $1, $ $2, $4, $2, $5, $46, $41, $9, $2,199, $2,190,

15 Future Value of a Single Cash Flow The future value (or sum) at i% p.a. of $P 0 today is the dollar value to which it grows at the end of time period n S n = P 0 (1 + i) n 15

16 Future Value of a Single Cash Flow The future value at r% p.a. of $P 0 today is the dollar value to which it grows at the end of time n FVIF r, n = $1(1 + r) n FVIF is short for Future Value Interest Factor 16

17 Future Value of a Single Cash Flow Example: You decide to invest $1,000 for different time periods. What is the future value of this $1,000 in 5, 20 and 100 years at an interest rate of (a) 4% and (b) 6%? At i = 4% p.a. S 5 = 1000 (1.04) 5 = $1,217 S 20 = 1000 (1.04) 20 = $2,191 S 100 = 1000 (1.04) 100 = $50,505 At i = 6% p.a. S 5 = 1000 (1.06) 5 = $1,338 S 20 = 1000 (1.06) 20 = $3,207 S 100 = 1000 (1.06) 100 = $339,302 17

18 Future Value of a Single Cash Flow 18

19 Future Value of a Single Cash Flow The future value of a cash flow depends on the following factors: The time period, n Future value increases as n increases The interest rate, i Future value increases as i increases The method of calculating interest Future value increases as the compounding interval increases (more on this later). 19

20 Present Value of a Single Cash Flow The present value (P 0 ) at i% p.a. of $S n at the end of time n is the amount which invested today would grow to $S n in time n P 0 = S n / (1 + i) n = S n (1 + i) -n 20

21 Present Value of a Single Cash Flow The present value (PV) at r% p.a. of $1 at the end of time n is the amount which invested now would grow to $1 in time n PVIF r, n = $1/(1 + r) n = $1(1 + r) -n PVIF is the short for Present Value Interest Factor Note: PVIF r, n = 1/FVIF r, n 21

22 Present Value of a Single Cash Flow Example: If you needed $10,000 in (a) five years, (b) ten and (c) twenty years how much would you need to save and invest today if the interest rates were (a) 4% and (b) 6%? The present value of $10,000 in five years At 4% p.a., P 0 = 10000/(1.04) 5 = $8, At 6% p.a., P 0 = 10000/(1.06) 5 = $7, The present value of $10,000 in ten years At 4% p.a., P 0 = 10000/(1.04) 10 = $6, At 6% p.a., P 0 = 10000/(1.06) 10 = $5, The present value of $10,000 in twenty years At 4% p.a., P 0 = 10000/(1.04) 20 = $4, At 6% p.a., P 0 = 10000/(1.06) 20 = $3,

23 Present Value of a Single Cash Flow 23

24 Factors Influencing Present and Future Values The present and future values of a cash flow depend on the following factors The time period, n Future value increases as n increases Present value decreases as n increases The interest rate, i Future value increases as i increases Present value decreases as i increases The method of calculating interest Future value increases as the compounding interval increases Present value decreases as the compounding interval increases 24

25 Net Present Value Net Present Value (NPV) is defined as the present value (PV) of cash inflows minus the present value of cash outflows NPV = PV(Cash inflows) - PV(Cash outflows) Class Exercise 1: Your company is considering investing $900,000 in a project that is expected to yield the following net cash flows over its fouryear life. Assume that the company uses an interest rate of 12% p.a. to evaluate its investments. Should your company make this investment? 25

26 Answer to Class Exercise 1 Need to look at whether PV(Cash Inflows) > PV(Cash Outflows) Net Present Value = PV(Cash Inflows) - PV(Cash Outflows) Accept project if NPV > 0 Reject project if NPV < 0 Point of indifference where NPV = 0 PV(Cash Inflows) = / / / / = $1,097,402 PV(Cash Outflows) = $900,000 Net Present Value, NPV = $1,097,402 - $900,000 = $197,402 > 0 26

27 Valuing Unequal Cash Flows Class Exercise 2: You decide to invest $1,000 at the end of year 1 and then an additional $1,000 at the end of every year for five years. What is the future value of these cash flows at the end of five years? What equivalent lump-sum amount could you invest today to get this future amount? Assume an interest rate of 10% p.a. 27

28 Answer to Class Exercise 2 To get the total future (present) value of different cash flows occurring at different time periods compute their individual future (present) values and then add across Future value of cash flows at the end of five years FV 5 = 1000 (1.10) (1.10) (1.10) (1.10) FV 5 = $17, Equivalent single amount that could be invested today to get this future amount PV 0 = 1000/(1.10) /(1.10) /(1.10) /(1.10) /(1.10) 5 = $10, or equivalently PV 0 = FV 5 / (1.10) 5 = /(1.10) 5 = $10,

29 Key Concepts Managers should aim to maximize the value of the firm by taking on profitable investments. This results in maximum shareholder wealth. Money has time value because of compounded interest In simple interest, the value of a cash flow is calculated without including any accrued interest to the principal In compounded interest, the value of a cash flow is calculated based on the principal and interest accrued The present and future values of a cash flow depend on the time period, interest rate, and method of calculating interest 29

30 Key Relationships Future value (Simple interest) S n = P 0 (1 + n i) Present value (Simple interest) P 0 = S n / (1 + n i) Future value (or sum) of $P 0 S n = P 0 (1 + i) n Present value of $Sn P 0 = S n / (1 + i) n = S n (1 + i) -n 30

31 Introduction to Financial Economics Objectives Compute an unknown interest rate and time period Define and compute effective interest rates Compute the present value perpetuities Compute the present and future values of ordinary annuities Compute the present and future values of annuities due Applications using financial mathematics 31

32 Unknown Interest Rate or Time Period Example: You invest $10,000 for a five year period. What interest rate do you need to earn for the funds to double in that time period? If you invest $10,000 at an interest rate of 10% p.a. how long will it take for these funds to double in value? In the first case we have an unknown interest rate, i P 0 = $10,000, S 5 = $20,000, n = 5, i =? 10,000 (1 + i) 5 = 20,000 So, (1 + i) 5 = 20,000/10,000 = 2 i = 21/5-1 = 14.9% Rule of 72: The approximate interest rate required to double your funds is given as: i 72/n i 72/5 = 14.4% 32

33 Unknown Interest Rate or Time Period In the second case we have an unknown time period, n P 0 = $10,000, S n = $20,000, i = 10%, n =? Rule of 72: The approximate time it will take for the funds to double is given as: n 72/i n 72/10 = 7.2 years (Note: use the interest rate in percentages) Alternatively, 10,000 (1.10) n = 20,000 So, (1.1) n = 20,000/10,000 = 2 Taking natural logs we have, n ln(1.1) = ln(2) So, n = / = 7.3 years 33

34 The Effective Interest Rate Interest may not always be earned or paid on an annual basis Example: Bank A pays an interest rate of 5% p.a. while Bank B pays an interest rate of 4.9% p.a. but with interest compounded monthly. Which bank s return is better? The effective interest rate (ie) is the annualized rate that takes account of compounding within the year ie = (1 + i/m) m - 1 where i/m is the per period interest rate As the compounding becomes more frequent, m approaches infinity, and in the limit (1 + i/m) m approaches ei Effective annual rate with continuous compounding: ie = ei- 1 (where e = ) 34

35 The Effective Interest Rate Example: Assume the stated interest rate is 5% p.a. What is the effective annual interest rate if interest is paid: (a) semiannually, (b) quarterly, (c) monthly, (d) daily, and (e) continuously? In some markets, (eg. the US) daily compounding is based on a 360 day year. What is the effective interest rate in this case? Effective interest rate, ie = (1 + i/m) m 1 Note: i = ie only when the compounding interval is one year (m = 1), otherwise ie will always exceed i The effective annual interest rate always rises with the compounding interval 35

36 The Effective Interest Rate Effective annual interest rates for different compounding intervals Semi-Annual: ie = ( /2) 2-1 = = % Quarterly: ie = ( /4) 4-1 = = % Monthly: ie = ( /12) 12-1 = = % Daily: ie = ( /365) = = % Continuous: ie = e = = % Daily (360 day basis): ie = ( /360) ie = = 5.2% 36

37 The Effective Interest Rate If the interest rate is i% p.a. but interest is paid m times a year, after n years $P 0 will have the following future value: S n = P 0 (1 + i/m) m n i/m = Per period interest rate m n = Total periods over which interest is compounded Example: Suppose your ancestor saved $1,000 one hundred years ago with interest compounded monthly. What would its (future) value be today at an interest rate of (a) 5% and (b) 7%? Here, m = 12 and m n = 1200 At 5% p.a., FV = 1000 ( /12) 1200 = $146,880 At 7% p.a., FV = 1000 ( /12) 1200 = $1,074,555! 37

38 Continuous Compounding The relationship between the present and future values when interest is compounded continuously is: FV = (PV)e r n and PV = (FV)e -r n ln(fv) = ln(pv) + rn r = (1/n) ln (FV/PV) Ln (FV/PV) is the log price relative Example: If FV in 1 year = $110, PV = $105, the interest rate is r = ln(110/105) = ln(1.0476) = or 4.65% 38

39 Continuous Compounding Class Exercise 1: Your great-grandmother invested $1,000, one hundred years ago earning continuously compounded interest a) How much is this amount worth today if the interest rate is 5% and 7%? b) What are the effective interest rates for the above stated rates when interest is continuously compounded? 39

40 Answer to Class Exercise 1 a) The (future) value of the $1,000 with continuous compounding is: At 5% p.a., FV = 1000(e 0.05 ) 100 = $148,413 At 7% p.a., FV = 1000(e 0.07 ) 100 = $1,096,633 Difference between monthly and continuous compounding at 7% is $10, $10, = $22,078 b) The effective interest rates are At 5%, re = e = = 5.127% p.a. At 7%, re = e = = 7.251% p.a. Note: The difference in the effective interest rate between daily and continuous compounding is very small 40

41 Valuing Perpetuities and Annuities A perpetuity is a equal, periodic cash flow that goes on forever The present value of a perpetuity is P 0 = C /(1+i) + C /(1+i) C /(1+i) n + C /(1+i) n+1 + P 0 = C [1/(1+i) + 1/(1+i) /(1+i) n + 1/(1+i) n+1 + ] As n approaches, [1/(1+i) + 1/(1+i) /(1+i)n + ] approaches 1/I So, the present value of a perpetuity, P 0 = C / i 41

42 Valuing Perpetuities and Annuities The present value of a perpetuity, P 0 = C / i A deferred perpetuity is a perpetuity that starts at some future date and then goes on forever The present value of a deferred perpetuity, P 0 = [C / i] / (1 + i) n 42

43 Valuing Perpetuities and Annuities Example: A prize guarantees you $1,000 per year forever with the first payment to be made at the end of year 1. How much would you sell the prize for if the interest rate were 10% p.a.? What would the perpetuity s value be if were deferred to year 4 (i.e., the first cash flow occurred at the end of year 4 and not year 1)? (See the time line in the previous slide) Present value of perpetuity, P 0 = 1000 / 0.10 = $10,000 Present value of deferred perpetuity P 3 = 1000 / 0.10 = $10,000 P 0 = [1/(1.1) 3 ] = $7,

44 Valuing Ordinary Annuities An ordinary annuity is a series of equal, periodic cash flows occurring at the end of each period and lasting for n periods Note: The first cash flow occurs at the end of period 1 and the last cash flow occurs at the end of period n Ordinary annuities can be valued as the difference between two perpetuities 44

45 Valuing Ordinary Annuities An n year annuity can be valued as the difference between two perpetuities, P 0 (OA) = P 1 - P 2 45

46 Valuing Ordinary Annuities The present value of a $C annuity can be obtained as the difference between an ordinary perpetuity and a deferred perpetuity P 0 (OA) = P 1 - P 2 P 0 (OA) = C / i - [C / i][1/(1 + i) n ] Simplifying the above equation, we get P 0 (OA) = [C / i][1-1/(1 + i) n ] = [C / i][1 - (1 + i) -n ] The future value of a $C annuity can be obtained by taking the future value of the above present value to time period n S n (OA) = [C / i][1-1/(1 + i) n ](1 + i) n Simplifying the above equation, we get S n (OA) = [C / i][(1 + i) n - 1] 46

47 Valuing Deferred Ordinary Annuities A deferred ordinary annuity is a series of equal, periodic cash flows occurring at the end of each period where the first cash flow occurs at a future date. 47

48 Valuing Deferred Ordinary Annuities Class Exercise 2: Suppose you invest $1,000 every year for (i) 10 years and (ii) 50 years earning an annual return of 10%. a) What is each investment s value at the point where you stop investing? b) What is the present worth of your investments in part (a)? c) What is the relationship between the future and present values calculated in parts (a) and (b)? d) What are the present and future values at the end of year 10 assuming that you only invest funds in years 6-10? (That is, no funds are invested during years 1-5) 48

49 Answer to Class Exercise 2 a) Future values at the end of 10 and 50 years In 10 years: S 10 = [1000/0.1][ ] = $15,937 In 50 years: S 50 = [1000/01][ ] = $1,163,909 b) Present values of the above investments Over 10 years: P 0 = [1000/0.1][( ] = $6,145 Over 50 years: P 0 = [1000/0.1][( ] = $9,915 c) If you had invested $9,915 for a 50 year period, it would be worth $1,163,909 at a 10% return p.a. 9915(1.1) 50 = $1,163,930 (rounding error) 49

50 Answer to Class Exercise 2 50

51 Valuing Annuities Due An annuity due is a series of equal, periodic cash flows occurring at the beginning of each period Note: The beginning of period t is the same as the end of period t-1. The overall effect is to move the annuity back one period on our standard (end of period) time line 51

52 Valuing Annuities Due The present value of a $C annuity due at i% p.a. is equivalent to the present value of an ordinary annuity compounded one additional period P 0 (AD) = [C / i][(1 - (1 + i) -n ][1 + i] The future value of a $C annuity due at i% p.a. is equivalent to compounding by one additional period the future value of an ordinary annuity S n (AD) = [C / i][(1 + i) n - 1][1 + i ] Class Exercise 3: In Class Exercise 2, suppose the $1,000 was invested every year at the beginning of each year. What are the future and present values of these investment earning a 10% p.a. return over (a) 10 years and (b) 50 years? 52

53 Answer to Class Exercise 3 Future values at the end of 10 and 50 years are now In 10 years: S 10 = [1000/0.1][ ][1.1] = $17,530 In 50 years: S 50 = [1000/01][ ][1.1] = $1,280,300 Present values of the above investments Over 10 years: P 0 = [1000/0.1][ ][1.1] = $6,759 Over 50 years: P 0 = [1000/0.1][ ][1.1] = $10,906 53

54 Valuing Perpetuities A perpetuity is an annuity that goes on forever and ever and... PV = C/r A perpetuity growing at g% p.a., compounded at r% p.a. has a present value of PV = C /(r - g) Notes: The first cash flow occurs at the end of year 1 The above relationship requires that r > g Class Exercise 4: A prize guarantees $1,000 per year forever with the first payment to be made at the end of year 1. a) What would you sell the prize to your lecturer for if the interest rate were 10% p.a.? b) If a prize guaranteed $1,000 per year forever (first cash flow to be paid in year 1) growing at 5% p.a., what would its worth be today if the interest rate were 10% p.a.? 54

55 Answer to Class Exercise 4 a) Present value = 1000 / 0.10 = $10,000 If the prize paid $1,000 per year for 60 years its present value would be PV = 1000(PVIFA 10, 60 ) = $9,967 b) Present value now = 1000/( ) = $20,000 Note: The difference of $10,000 in parts (a) and (b) is referred to as the present value of growth opportunities (more on this later) 55

56 Financial Mathematics Application Application: You have borrowed $20,000 from your bank with the loan to be repaid in equal annual installments. Your bank charges an annual interest rate of 10% p.a. with interest compounded annually a) What annual payment would you be making on this loan? b) Develop a loan amortization schedule for this loan. Then using this table obtain the following information (i) The principal balance outstanding at the end of year 1 (ii) The total interest paid in year 2 (iii) The total principal repaid in year 3 56

57 Financial Mathematics Application a) What annual payments would you be making during the next four years? Loan amount at any point = PV(Remaining payments) = Payment [ ]/0.1 = Payment (3.1699) Payment = 20000/ = $6,309 (rounded) b) The loan amortization schedule shows the interest paid, principal repaid and principal remaining over the loan s duration. It uses the following relationships: Interest paid = (Previous period s principal) (Interest rate) Principal repaid = Loan Payment - Interest paid Principal remaining = Previous period s principal Principal repaid 57

58 Financial Mathematics Application b) Loan amortization schedule 58

59 Key Concepts The effective interest rate is the annualized rate that takes account of compounding within the year Ordinary annuities are periodic, end-of-the-period cash flows Deferred ordinary annuities are periodic, end-of-the-period cash flows that start at a future date Annuities due are periodic, beginning-of-the-period cash flows The future value of an annuity is the sum of the future values of each cash flow compounded at the relevant interest rate The present value of an annuity is the sum of the present values of each cash flow discounted at the relevant interest rate 59

60 Key Concepts The total future (present) value of different cash flows occurring at different time periods equals the sum of their individual future (present) values The effective interest rate is the annualized rate that takes account of compounding within the year Ordinary annuities are periodic, end-of-the-period cash flows Deferred ordinary annuities are periodic, end-of-the-period cash flows that start at a future date Annuities due are periodic, beginning-of-the-period cash flows The future value of an annuity is the sum of the future values of each cash flow compounded at the relevant interest rate The present value of an annuity is the sum of the present values of each cash flow discounted at the relevant interest rate 60

61 Key Relationships Future value (or sum) of $P0 today: S n = P 0 (1 + i) n Present value of $Sn at time n: P0 = S n /(1 + i) n = S n (1 + i) -n Rule of 72: i 72/n Effective interest rate: ie = (1 + i/m) m 1 Present value of a perpetuity: C/I Present value of a deferred perpetuity: [C/i]/(1 + i) n Present value of ordinary annuity: P 0 (OA) = [C/i][1-1/(1 + i) n ] Future value of ordinary annuity: S n (OA) = [C/i][(1 + i) n - 1] Present value of an annuity due: P 0 (AD) = [C/i][(1 - (1 + i) -n ][1 + i] Future value of an annuity due: S n (AD) = [C/i][(1 + i) n - 1][1 + i ] 61